2021 Japan TST

Part of Japan TST

Subcontests

(3)

1

An international conference

n

be an integer greater than or equal to

2

n

people from each of

n

countries gather, a total of

n^2

people facing inward and surround the round table. For any two different people in the same country, their left neighbors belong to different countries. Find the maximum number of people whose left and right neighbors belong to the same country.

1

Wrapped FE

Find all functions

f: \mathbb{R} \to \mathbb{R}

f(x^2+xy^2+y^2)=2x^2f(y)+2xf(f(y))+f(-x^2-xy^2)+f(y^2)

holds for any real number

x

y

1

Combinatoric number theory

Find all integers

n

greater than or equal to

4

that satisfy the following conditions:
[*] Take an arbitrary convex

n

P

on the coordinate plane whose vertices are lattice points (points whose coordinates are both integers). There are a total of

n

triangles that share two sides with

P

S_1, S_2, \ldots, S_n

be their areas, and let

S

P

. Then, the greatest common divisor of

2S_1, 2S_2, \ldots, 2S_n

divides the integer

2S